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    Re: Classification of the methods for clearing the Lunar Distances
    From: Jan Kalivoda
    Date: 2003 Apr 11, 20:31 +0200

    Hello, All,
    I am sorry for my late answer. I spent some days at my mother, without a wire into the world.
    Thank you for your approval and comments. I will try to react to them one by one.
    To Bill Noyce:
    Yes, the effort to escape antilogs could promote the Borda's method. But the 
    Dunthone's formula requires three tables: of natural cosines, of "logarithmic 
    difference" and of logarithms of numbers. In Borda's method, one had to 
    browse through three tables, too: log cos, log sin and "logarithmic 
    difference". And the number of arithmetical operations was twelve in it; it 
    falls to number nine in the variant of this method that Herbert Prinz has 
    sent into the list in another reaction to my text. Dunthorne's method 
    required five operations.
    To George Huxtable:
    You are right, George. The amount of Mm is the difference between the Moon's 
    parallax in altitude and its refraction in altitude. This value is greatest 
    in 14 degrees of altitude: 55.5 arc-minutes, when the horizontal parallax of 
    the moon is greatest (61 arc-minutes) and 48 arc-minutes, when H.P. is 
    minimal (53 arc-minutes). Above and below 14 degrees this difference 
    decreases, as the refraction slims down more quickly with the growing 
    altitude than the parallax.
    To Bruce Stark:
    Your identification of the mysterious Thomson with the Canadian surveyor David 
    Thompson is very interesting. But according to the last edition of 
    Encyclopedia Britannica, this David Thompson died at 1857 and there was a 
    foreword in the edition of the Thomson's tables from 1845 (written by Boulter 
    J. Bell), which speaks about Thomson as about a dead man. And G.Coleman (the 
    editor of Norie's "Epitome of practical Navigation" after Norie's death) said 
    before 1849 that he was in contact with "Captain Thomson". And thirdly, the 
    Thomson's tables had been edited by the publisher W. Allen & Co. from the 
    first edition, exactly by the same publisher , who edited the 67th edition 
    from 1880 you have in your library. I doubt, whether the publisher's 
    mystification replacing the surveyor Thompson for a captain Thomson could 
    survive such long decades? Maybe it could?
    Thank you for that information about the 67th edition - it is fascinating to 
    see such a long life for a set of nautical tables; only Inman, Norie and 
    Bowditch can beat Thomson, isn't it?
    As for the year 1824 of the first edition, you are right, I have checked it 
    now in the online catalogue of British Library. My source was in error and I 
    took it over wrongly.
    I am very interested in your tables for clearing L.D.'s, even more after 
    reading praise of them in the list.
    To Arthur Pearson:
    Thank you for your assessment, Arthur. But don't ask graphics from me, please! 
    My teacher of drawing would be very angry, as he was in my childhood. For 
    that matter, the picture No.6 on your nice website (in PDF-file) is exactly 
    the proper one for illustrating my text about approximative and rigorous 
    methods; only the scale can be enlarged and the symbols M,m,S,s added. It 
    would suffice.
    I will try to send a more detailed description of approximative formulas for 
    clearing the L.D.'s in the next hours or days, as you wanted.
    To Herbert Prinz and Fred Hebart:
    Yes, I had stressed that my analogies between the methods of clearing the 
    L.D.'s and resolving the LOP's are only very rough. The intercept method for 
    Marcq St. Hilaire problem is an approximative solution in the core, the 
    rigorous one would be to resolve the system of two circle equations on the 
    surface of the Earth's sphere (even ellipsoid!). But Fred said it for me - 
    the cosine-haversine equation is strictly deduced from basic trigonometric 
    formulas for spherical triangles as in the rigorous methods for clearing the 
    L.D.'s. This was my argument in comparing these two sets of procedures.
    As for Chauvenet, it was my greatest blunder. I saw him cited in an old German 
    article as a French astronomer and under the French title of his work - the 
    French title, the "French" name, the French attribution - I didn't verify it 
    once more. I blush and thank you for the correction.
    Thank you for all comments, once more.
    Jan Kalivoda

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